Topological Superconductors and Category...

Topological Superconductors and Category Theory

Andrei Bernevig1 and Titus Neupert2

1Department of Physics, Princeton University,

Princeton, New Jersey 08544, USA

2Princeton Center for Theoretical Science,

Princeton University, Princeton, New Jersey 08544, USA

(Dated: September 18, 2015)

Abstract

We give a pedagogical introduction to topologically ordered states of matter, with the aim of

familiarizing the reader with their axiomatic topological quantum field theory description. We in-

troduce basic noninteracting topological phases of matter protected by symmetries, including the

Su-Schrieffer-Heeger model and the one-dimensional p-wave superconductor. The defining proper-

ties of topologically ordered states are illustrated explicitly using the toric code and – on a more

abstract level – Kitaev’s 16-fold classification of two-dimensional topological superconductors. Sub-

sequently, we present a short review of category theory as an axiomatic description of topological

order in two-dimensions. Equipped with this structure, we revisit Kitaev’s 16-fold way.

These lectures were in parts held at:

• Les Houches Summer School “Topological Aspects of Condensed Matter Physics”, 4–29 Au-

gust 2014, École de Physique des Houches, Les Houches, France

• XVIII Training Course in the Physics of Strongly Correlated Systems, 6–17 October 2014,

International Institute for Advanced Scientific Studies, Vietri sul Mare, Italy

• 7th School on Mathematical Physics “Topological Quantum Matter: From Theory to Appli-

cations”, 25–29 May 2015, Universidad de los Andes, Bogotá, Colombia

1

CONTENTS

I. Introduction to topological phases in condensed matter 4

A. The notion of topology 4

B. Classification of noninteracting fermion Hamiltonians: The 10-fold way 7

1. Classification with respect to time-reversal and particle-hole symmetry 7

2. Flatband Hamiltonians and homotopy groups 9

3. Topological invariants 12

C. The Su-Schrieffer-Heeger model 16

D. The one-dimensional p-wave superconductor 18

E. Reduction of the 10-fold way classification by interactions: Z→ Z8 in class BDI 22

II. Examples of topological order 26

A. The toric code 26

1. Ground states 28

2. Topological excitations 31

B. The two-dimensional p-wave superconductor 34

1. Argument for the existence of Majorana bound states on vortices 40

2. Bound states on vortices in two-dimensional chiral p-wave superconductors 42

3. Non-Abelian statistics of vortices in chiral p-wave superconductors 43

4. The 16-fold way 46

III. Category theory 49

A. Fusion Category 49

1. Diagrammatics 51

2. F-moves and the pentagon equation 52

3. Gauge freedom and its fixing 54

4. Quantum dimensions and Frobenius Schur indicators 55

5. Examples 57

B. Braiding Category 58

1. Topological spin 59

2. Ribbon equation 61

3. Vafa’s Theorem 62

2

C. Modular matrices 62

1. The S matrix 63

2. Verlinde Formula 64

3. Obstruction for theories with multiplicities 66

4. The T matrix 67

D. Examples: The 16-fold way revisited 67

1. Case: C(1) odd 67

2. Case C(1) = 2 mod 4 68

3. Case C(1) = 0 mod 4 69

Acknowledgements 70

References 70

3

I. INTRODUCTION TO TOPOLOGICAL PHASES IN CONDENSED MATTER

A. The notion of topology

In these lectures we will learn how to categorize and characterize some phases of matter

that have topological attributes. A topological property of a phase, such as boundary modes

(in an open geometry), topological response functions, or the character of its excitations, is

described by a set of quantized numbers, related to so-called topological invariants of the

phase. The quantization immediately implies that topological properties are universal (they

can be used to label the topological phase) and in some sense protected, because they cannot

change smoothly when infinitesimal perturbations are added. Topological properties, in the

sense that we want to discuss them here, can only be defined for

• spectrally gapped ground states on a manifold without boundary of

• local Hamiltonians at

• zero temperature.

The spectral gap allows to define an equivalence class of states, i.e., a phase, with the help

of the adiabatic theorem. Two gapped ground states are in the same phase if there exists

an adiabatic interpolation between their respective Hamiltonians, such that the spectral gap

above the ground state as well as the locality is preserved for all Hamiltonians along the

interpolation.

Often it is useful to further modify these rules to define topological phases that are

subject to symmetry constraints. We refer to topological states as being protected/enriched

by a symmetry group G, if the Hamiltonian has a symmetry G and only G-preserving

interpolations are allowed. Since the G-preserving interpolations are a subset of all local

interpolations, it is clear that symmetries make a topological classification of Hamiltonians

more refined.

The locality of a Hamiltonian is required to guarantee the quantization of topological

response functions and to distinguish topological characterizations depending on the di-

mensionality of space. If we were not to impose locality, any system could in essence be

zero-dimensional and there would be no notion of boundary states (which are localized over

short distances) or point-like and line-like excitations etc.

4

Equipped with this definition of a topological phase, the exploration of topological states

of matter above all poses a classification problem. We would like to know how many phases

of quantum systems exists, that can be distinguished by their topological properties. We

would like to obtain such a classification while imposing any symmetry G that is physically

relevant, such as time-reversal symmetry, space-group or point-group symmetries of a crystal,

particle-number conservation etc. To identify the right mathematical tools that allow for

such a classification and to guarantee its completeness is a subject of ongoing research.

Here, we shall focus on aspects of this classification problem, which are well established and

understood.

Most fundamental is a distinction between two types of topological states of matter:

Those with intrinsic (long-range entangled) topological order 1 and those without. This

notion is also core to the structure of these lecture notes. In this Section, we only discuss

phases without intrinsic topological order, while the ensuing two Sections are devoted to

states with intrinsic topological order. A definition of intrinsic topological order can be

based on several equivalent characterizations of such a phase, of which we give three:

• Topological ground state degeneracy: On a manifold without boundary, the degeneracyof gapped topologically degenerate ground states depends on the topological properties

of the manifold. There are no topologically degenerate ground states if the system is

defined on a sphere. The matrix elements of any local operator taken between two

distinct topologically degenerate ground states vanishes.

• Fractionalized excitations: There exist low-energy excitations which are point-like[in two dimensions (2D) or above] or line-like [in three dimensions (3D) or above].

These excitations carry a fractional quantum numbers as compared to the microscopic

degrees of freedom that enter the Hamiltonian (for example, a fractional charge), are

deconfined and dynamical (i.e, free to move in the low-energy excited states).

• Topological entanglement entropy: The entanglement entropy between two parts of asystem that is in a gapped zero-temperature ground state typically scales with the size

of the line/surface that separates the two regions (“area-law entanglement”). Topo-

logically ordered, long-range entangled states have a universal subleading correction

to this scaling that is characteristic for the type of topological order.

5

(Note that these statements, as many universal properties we discuss, are only strictly

true in the thermodynamic limit of infinite system size. For example, in a finite system,

the ground state degeneracy is lifted by an amount that scales exponential in the system

size.) As fractionalized excitations in the above sense may only exist in two or higher

dimensions, intrinsic topological order cannot be found in one-dimensional (1D) phases of

matter. Further, for intrinsic topological order to occur, interactions are needed in the

system.

Examples of topologically nontrivial phases (both with and without intrinsic topological

order) exist in absence of any symmetry. However, most of the phases without intrinsic

topological order belong to the so-called symmetry protected topological (SPT) phases. In

these cases, the topology is protected by a symmetry. These phases almost always possess

topologically protected boundary modes when defined on a manifold with boundary, except

if the boundary itself breaks the protecting symmetry (as could be the case with inversion

symmetry, for example).

In contrast, phases with intrinsic topological order are not necessarily equipped with

boundary modes, even if the boundary of the manifold preserves the defining symmetries of

the phase. If the definition of a phase with intrinsic topological order relies on symmetries,

it is named symmetry enriched topological phase (SET).

An alternative characterization of topological properties of a phase uses the entanglement

between different subsystems. While we opt not to touch upon this concept here, we want

to make contact to the ensuing terminology: All phases with intrinsic topological order are

called long-range entangled (LRE). The term short-range entangled (SRE) phase is often

used synonymously with “no intrinsic topological order”. (Some authors count 2D phases

with nonvanishing thermal Hall conductivity, such as the p + ip superconductors, but no

intrinsic topological order unless gauged, as LRE.)

In these lecture notes, we will encounter two classifications of a subset of topological

phases. The following Subsection introduces the complete classification of non-interacting

fermionic Hamiltonians with certain symmetries (which have no intrinsic topological order).

Section III is concerned with the unified description of 2D phases with intrinsic topological

order in absence of any symmetries.

6

B. Classification of noninteracting fermion Hamiltonians: The 10-fold way

We have stated that SPT order in SRE states manifests itself via the presence of gapless

boundary states in an open geometry. In fact, there exists a intimate connection between

the topological character of the gapped bulk state and its boundary modes. The latter are

protected against local perturbations on the boundary that (i) preserve the bulk symmetry

and (ii) induce no intrinsic topological order or spontaneous symmetry breaking in the

boundary modes. This bulk-boundary correspondence can be used to classify SPT phases.

Two short-range entangled phases with the same symmetries belong to a different topological

class, if the interface between the two phases hosts a state in the bulk gap and this state

cannot be moved into the continuum of excited states by any local perturbation that obey

(i) and (ii). Equivalently, to change the topological attribute of a gapped bulk state via any

smooth changes in the Hamiltonian, the bulk energy gap has to close and reopen.

In Ref. 2 Schnyder et al. use this bulk-boundary correspondence to classify all nonin-

teracting fermionic Hamiltonians. For the topological phases that they discuss, two funda-

mental symmetries, particle-hole symmetry (PHS) and time-reversal symmetry (TRS), are

considered. In the following, we will review the essential results of this classification.2–4

1. Classification with respect to time-reversal and particle-hole symmetry

Symmetries in quantum mechanics are operators that have to preserve the absolute value

of the scalar product of any two vectors in the Hilbert space. They can thus be either unitary

operators, preserving the scalar product, or antiunitary operators, turning the scalar product

into its complex conjugate (up to a phase). For a unitary operator to be a symmetry of a

given Hamiltonian H, the operator has to commute with H. Consequently, the Hamiltonian

can be block diagonalized, where each block acts on one eigenspace of the unitary symmetry.

If H has a unitary symmetry, we block-diagonalize it and then consider the topological

properties of each block individually. This way, we do not have to include unitary symmetries

(except for the product of TRS and PHS and the omnipresent particle number conservation)

in the further considerations, as we will not focus on the burgeoning field of crystalline

topological insulators.

A fundamental antiunitary operator in quantum mechanics is the reversal of time T . Let

7

us begin by recalling its elementary properties. If a given Hamiltonian H is TRS, that is,

T HT −1 = +H, (1a)

the time-evolution operator at time t should be mapped to the time-evolution operator at

−t by the operator T

T e−itHT −1 = e−T iT −1tH

= e−i(−t)H .(1b)

We conclude that the reversal of time is indeed an antiunitary operator T iT −1 = −i. Itcan be represented as T = TK, where K denotes complex conjugation and T is a unitaryoperator. Applying the reversal of time twice on any state must return the same state up

to an overall phase factor eiφ

eiφ!

= T 2 = T (TT)−1 ⇒ T = eiφTT, TT = eiφT. (1c)

Inserting the two last equations into one another, one obtains T = e2iφT , i.e., e2iφ has to

equal +1. We conclude that the time-reversal operator either squares to +1 or to −1

T 2 = +1, T 2 = −1. (1d)

The second fundamental antiunitary symmetry considered here is charge conjugation P .Its most important incarnation in solid state physics is found in the theory of supercon-

ductivity. In an Andreev reflection process, an electron-like quasi particle that enters a

superconductor is reflected as a hole-like quasi particle. The charge difference between inci-

dent and reflected state is accounted for by adding one Cooper pair to the superconducting

condensate. In the mean-field theory of superconductivity, the energies of the electron-like

state and the hole-like state are equal in magnitude and have opposite sign, giving rise to

the PHS. In this case, rather than being a fundamental physical symmetry of the system

like TRS is, PHS emerges due to a redundancy in the mean-field description. We define a

(single-particle) Hamiltonian H to be PHS if

PHP−1 = +H. (1e)

In order to also reverse the sign of charge, P has to turn the minimal coupling p − ieAinto p + ieA, where p is the momentum operator and A is the electromagnetic gauge

8

potential. This is achieved by demanding P iP−1 = −i. We conclude that P is indeed anantiunitary operator that can be decomposed as P = PK, where P is a unitary operator. Asa consequence, the reasoning of Eq. (1c) also applies to P and we conclude that the chargeconjugation operator either squares to +1 or to −1

P2 = +1, P2 = −1. (1f)

In the case where the operators T and P are both symmetries of H, their product isalso a symmetry of H. We call this product chiral transformation C := T P . It is a unitaryoperator. The Hamiltonian H transforms under the chiral symmetry as

CHC−1 = +H. (1g)

(It is important to note that both P and C anticommute rather than commute with thesingle-particle first-quantized HamiltonianHα,α′ that we will introduce below.) Observe thata Hamiltonian can have a chiral symmetry, even if it possesses neither of PHS and TRS. We

can now enumerate all combinations of the symmetries P , T , and C that a Hamiltonian canobey, accounting for the different signs of T 2 and P2. There are in total ten such symmetryclasses, listed in Tab. I. The main result of Schnyder et al. in Ref. 2 is to establish how

many distinct phases with protected edge modes exist on the (d− 1)-dimensional boundaryof a phase in d dimensions. We find three possible cases: If there is only one (topologically

trivial) phase, the entry ∅ is found in Tab. I. If there are exactly two distinct phases (one

trivial and one topological phase), Z2 is listed. Finally, if there exists a distinct topological

phase for every integer, Z is listed.

2. Flatband Hamiltonians and homotopy groups

There are several approaches to obtain the entries Z2 and Z in Tab. I. For one, the

theory of Anderson localization can be employed to determine in which spatial dimensions

boundaries can host localization-protected states (the topological surface states) under a

given symmetry. This was done by Schnyder et al. in Ref. 2. Kitaev, on the other hand,

derived the table using the algebraic structure of Clifford algebras in the various dimensions

and symmetry classes.4 In mathematics, this goes under the name K-theory.

Here, we want to give a flavor of the mathematical structure behind the table by con-

sidering two examples. To keep matters simple, we shall restrict ourselves to the situation

9

TABLE I. Symmetry classes of noninteracting fermionic Hamiltonians from Refs. 3 and 4. The

columns contain from left to right: Cartan’s name for the symmetry class; the square of the time

reversal operator, the particle-hole operator, and the chiral operator (∅ means the symmetry is

not present); the group of topological phases that a Hamiltonian with the respective symmetry

can belong to for the dimensions d = 1, · · · , 8 of space. The first two rows are called “complex

classes”, while the lower eight rows are the “real classes”. The homotopy groups of the former show

a periodicity with period 2 in d, while those of the latter have a period 8 in d (Bott periodicity).

T 2 P2 C2 d 1 2 3 4 5 6 7 8

A ∅ ∅ ∅ ∅ Z ∅ Z ∅ Z ∅ Z

AIII ∅ ∅ + Z ∅ Z ∅ Z ∅ Z ∅

AII − ∅ ∅ ∅ Z2 Z2 Z ∅ ∅ ∅ Z

DIII − + + Z2 Z2 Z ∅ ∅ ∅ Z ∅

D ∅ + ∅ Z2 Z ∅ ∅ ∅ Z ∅ Z2

BDI + + + Z ∅ ∅ ∅ Z ∅ Z2 Z2

AI + ∅ ∅ ∅ ∅ ∅ Z ∅ Z2 Z2 Z

CI + − + ∅ ∅ Z ∅ Z2 Z2 Z ∅

C ∅ − ∅ ∅ Z ∅ Z2 Z2 Z ∅ ∅

CII − − + Z ∅ Z2 Z2 Z ∅ ∅ ∅

where the system is translationally invariant and periodic boundary conditions are imposed.

In second quantization, the Hamiltonian H has the Bloch representation

H =

∫ddkψ†α(k)Hα,α′(k)ψα′(k), (2a)

where ψ†α(k) creates a fermion of flavor α = 1, · · · , N at momentum k in the Brillouin zone(BZ) and the summation over α and α′ is implicit. The flavor index may represent orbital,

spin, or sublattice degrees of freedom. Energy bands are obtained by diagonalizing the

N ×N matrix H(k) at every momentum k ∈ BZ with the aid of a unitary transformationU(k)

U †(k)H(k)U(k) = diag[εm+n(k), · · · , εn+1(k), εn(k), · · · , ε1(k)

], (2b)

where the energies are arranged in descending order on the righthand side and n,m ∈ Z

10

such that n+m = N . So as to start from an insulating ground state, we assume that there

exists an energy gap between the bands n and n+ 1 and that the chemical potential µ lies

in this gap

εn(k) < µ < εn+1(k), ∀k ∈ BZ. (3)

The presence of the gap allows us to adiabatically deform the Bloch Hamiltonian H(k) tothe flatband Hamiltonian

Q(k) := U(k)

11m 0

0 −11n

U †(k) (4a)

that assigns the energy −1 and +1 to all states in the bands below and above the gap, respec-tively. This deformation preserves the eigenstates, but removes the nonuniversal information

about energy bands from the Hamiltonian.

In other words, the degenerate eigenspaces of the eigenvalues ±1 of Q(k) reflect the par-titioning of the single-particle Hilbert space introduced by the spectral gap in the spectrum

of H(k). The degeneracy of its eigenspaces equips Q(k) with an extra U(n)× U(m) gaugesymmetry: While the (n+m)× (n+m) matrix U(k) of Bloch eigenvectors that diagonalizesQ(k) is an element of U(n + m) for every k ∈ BZ, we are free to change the basis for itslower and upper bands by a U(n) and U(m) transformation, respectively. Hence Q(k) is anelement of the space C0 := U(n+m)/[U(n)× U(m)] defining a map

Q : BZ→ C0 . (5)

The group of topologically distinct maps Q, or, equivalently, the number of topologicallydistinct Hamiltonians H, is given by the homotopy group

πd (C0) (6)

for any dimension d of the BZ. (The homotopy group is the group of equivalence classes of

maps from the d-dimensional sphere to a target space, in this case C0. Even though the BZ

is a d-dimensional torus, it turns out that this difference between torus and sphere does not

affect the classification as discussed here.)

For example, in d = 2 we have π2 (C0) = Z. A physical example of a family of Hamil-

tonians that exhausts the topological sectors of this group is found in the integer quantum

Hall effect. The incompressible ground state with r ∈ N filled Landau levels is topologically

11

distinct from the ground state with N 3 r′ 6= r filled Landau levels. Two different patches ofspace with r and r′ filled Landau levels have |r− r′| gapless edge modes running at their in-terface, reflecting the bulk-boundary correspondence of the topological phases. In contrast,

π3 (C0) = Z1 renders all noninteracting fermionic Hamiltonians in 3D space topologically

equivalent to the vacuum, if no further symmetries besides the U(1) charge conservation are

imposed.

As a second example, let us discuss a Hamiltonian that has only chiral symmetry and

hence belongs to the symmetry class AIII. The chiral symmetry implies a spectral symmetry

of H(k). If gapped, H(k) must have an even number of bands N = 2n, n ∈ Z. Whenrepresented in the eigenbasis of the chiral symmetry operator C, the spectrally flattened

Hamiltonian Q(k) and the chiral symmetry operator have the representations

Q(k) =

0 q(k)q†(k) 0

, C =

11n 0

0 −11n

, (7a)

respectively. From Q(k)2 = 1, one concludes that q(k) can be an arbitrary unitary matrix.We are thus led to consider the homotopy group πd(C1) of the mapping

q : BZ→ C1 = U(n). (7b)

For example, in d = 1 spatial dimensions π3(C1) = Z. A tight-binding model with non-trivial

topology that belongs to this symmetry class will be discussed in Sec. I C.

With these examples, we have discussed the two complex classes A and AIII. In the real

classes, which have at least one antiunitary symmetry, it is harder to obtain the constraints

on the spectrally flattened Hamiltonian Q(k). The origin for this complication is that theantiunitary operators representing time-reversal and particle-hole symmetry relates Q(k)and Q(−k) rather than acting locally in momentum space.

3. Topological invariants

Given a gapped noninteracting fermionic Hamiltonian with certain symmetry properties

in d-dimensional space, one can use Tab. I to conclude whether the system can potentially

be in a topological phase. However, to understand in which topological sector the system is,

we have to do more work. To obtain this information, one computes topological invariants

12

or topological quantum numbers of the ground state. Such invariants are automatically

numbers in the group of possible topological phases (Z or Z2). For many of them, a variety

of different-looking but equivalent representations are known.

To give concrete examples, we shall discuss the invariants for all Z topological phases

found in Tab. I. These are called Chern numbers in the symmetry classes without chiral

symmetry and winding numbers in the classes with chiral symmetry.

In physics, topological attributes refer to global properties of a physical system that is

made out of local degrees of freedom and might only have local, i.e., short-ranged, correla-

tions. The distinction between global and local properties parallels the distinction between

topology and geometry in mathematics, where the former refers to global structure, while

the latter refers to local structure of objects. In differential geometry, a bridge between

topology and geometry is given by the Gauss-Bonnet theorem. It states that for compact

2D Riemannian manifolds M without boundary, the integral over the Gaussian curvature

F (x) of the manifold is (i) integer and (ii) a topological invariant

2(1− g) = 12π

∫

M

d2xF (x). (8)

Here, g is the genus of M , e.g., g = 0 for a 2D sphere and g = 1 for a 2D torus. The Gaussian

curvature F (x) can be defined as follows. Attach to every point on M the tangential plane,

a 2D vector space. Take some vector from the tangential plane at a given point on M and

parallel transport it around an infinitesimal closed loop on M . The angle mismatch of the

vector before and after the transport is proportional to the Gaussian curvature enclosed in

the loop.

In the physical systems that we want to describe, the manifold M is the BZ and the

analogue of the tangent plane on M is a space spanned by the Bloch states of the occupied

bands at a given momentum k ∈ BZ. The Gaussian curvature of differential geometry isnow generalized to a curvature form, called Berry curvature F. In our case, it is given by

an n× n matrix of differential forms that is defined via the Berry connection A as

F := Fij(k) dki ∧ dkj (9a)

Fij(k) := ∂iAj(k)− ∂jAi (k) + [Ai (k), Aj(k)], i, j = 1, · · · , d, (9b)

A := Ai (k) dki , (9c)

A(ab)i (k) :=

N∑

α=1

U †aα(k)∂iUαb(k), a, b = 1, · · · , n, i = 1, · · · , d. (9d)

13

(Two different conventions for the Berry connection are commonly used: Either it is purely

real or purely imaginary. Here we choose the latter option.) The unitary transformation

U(k) that diagonalizes the Hamiltonian was defined in Eq. (2b), both Ai (k) and Fij(k)

are n × n matrices, we write ∂i ≡ ∂/∂ki and the sum over repeated spatial coordinatecomponents i, j is implicit.

Under a local U(n) gauge transformation in momentum space that acts on the states of

the lower bands and is parametrized by the n× n matrix G(k)

Uαa(k) −→ Uαb(k)Gba(k), α = 1, · · · , N, a = 1, · · · , n, (10a)

the Berry connection A changes as

A −→ G†AG+G†dG, (10b)

while the Berry curvature F changes covariantly

F −→ G†FG, (10c)

leaving its trace invariant.

a. Chern numbers For the spatial dimension d = 2, the generalization of the Gauss-

Bonnet theorem (8) in algebraic topology was found by Chern to be

2C(1) :=i

2π

∫

BZ

tr F

= 2i

2π

∫

BZ

d2k trF12.

(11)

This defines a gauge-invariant quantity, the first Chern number C(1). Remarkably, C(1) can

only take integer values. In order to obtain a topological invariant for any even dimension

d = 2s of space, we can use the s-th power of the local Berry curvature form F (using the

wedge product) to build a gauge invariant d-form that can be integrated over the BZ to obtain

scalar. Upon taking the trace, this scalar is invariant under the gauge transformation (10a)

and defines the s-th Chern number

2C(s) :=1

s!

(i

2π

)s ∫

BZ

tr [Fs] , (12)

where Fs = F ∧ · · · ∧ F. As with the case s = 1 that we have exemplified above, C(s) isinteger for any s = 1, 2, · · · .

14

From inspection of Tab. I we see that symmetry classes without chiral symmetry may

have integer topological invariants Z only when the dimension d of space is even. In fact, all

the integer invariants of these classes are given by the Chern number C(d/2) of the respective

dimension.

b. Winding numbers Let us now consider systems with chiral symmetry C. To con-

struct their topological invariants as a natural extension of the above, we consider a different

representation of the Chern numbers C(s). In terms of the flatland projector Hamiltonian

Q(k) that was defined in Eq. (4a), we can write

C(s) ∝ εi1···id∫

BZ

ddk tr[Q(k)∂i1Q(k) · · · ∂idQ(k)

], d = 2s. (13)

The form of Eq. (13) allows to interpret C(s) as the winding number of the unitary trans-

formation Q(k) over the compact BZ. One verifies that C(s) = 0 for symmetry classes withchiral symmetry by inserting CC† at some point in the expression and anticommuting C

with all Q, using the cyclicity of the trace. After 2s + 1 anticommutations, we are backto the original expression up to an overall minus sign and found C(s) = −C(s). Hence, allsystems with chiral symmetry have vanishing Chern numbers.

In odd dimensions of space, we can define an alternative topological invariant for systems

with chiral symmetry by modifying Eq. (13) and using the chiral operator C

W(s) :=(−1)ss!

2(2s+ 1)!

(i

2π

)s+1εi1···id

∫

BZ

ddk tr[CQ(k)∂i1Q(k) · · · ∂idQ(k)

]

=(−1)ss!

(2s+ 1)!

(i

2π

)s+1εi1···id

∫

BZ

ddk tr[q†(k)∂i1q(k)∂i2q

†(k) · · · ∂idq(k)], d = 2s+ 1.

(14)

Upon anticommuting the chiral operator C once with all matrices Q and using the cyclicityof the trace, one finds that the expression for W(s) vanishes for even dimensions. The second

line of Eq. (14) allows to interpret W(s) as the winding number of the unitary off-diagonal

part q(k) of the chiral Hamiltonian that was defined in Eq. (7a). With Eq. (14) we have

given topological invariants for all entries Z in odd dimensions d in Tab. I.

In summary, we have now given explicit formulas for the topological invariants for all

entries Z in Tab. I for systems with translational invariance. It is important to remember

that the classification of Tab. I is restricted to systems without interactions. If interactions

15

are allowed, that neither spontaneously nor explicitly break the defining symmetry of a

symmetry class, one of two things can happen: i) Two phases which are distinguished

by a noninteracting invariant like W(0) might, sometimes but not always, be connected

adiabatically (i.e., without a closing of the spectral gap) by turning on strong interactions.

ii) Interactions can enrich the classification of Tab. I by inducing new phases with topological

response functions that are distinct from those of the noninteracting phases. We will given

an example for the scenario i) in Sec I E.

Besides, interactions can strongly modify the topological boundary modes of the nonin-

teracting systems to the extend that they can be gapped without breaking the protective

symmetries, but at the expense of introducing topological order on the boundary.

C. The Su-Schrieffer-Heeger model

The first example of a topological band insulator that we consider here is also the simplest:

The Su-Schrieffer-Heeger model5 describes a 1D chain of atoms with one (spinless) electronic

orbital each at half filling. The model was originally proposed to describe the electronic

structure of polyacetylene. This 1D organic molecule features a Peierls instability by which

the hopping integral between consecutive sites is alternating between strong and weak.

This enlarges the unit cell to contain two sites A and B. The second-quantized mean-field

Hamiltonian reads

H = tN∑

i=1

[(1− δ)c†A,icB,i + (1 + δ)c†B,icA,i+1 + h.c.

]. (15)

Here, c†A,i and c†B,i create an electron in the i-th unit cell on sublattice A and B, respec-

tively. If we identify i = N + 1 ≡ 1, periodic boundary conditions are implemented. Thecorresponding Bloch Hamiltonian

H = t∑

k∈BZ

∑

α=A,B

c†α,khαβ,kcβ,k (16a)

hk =

0 (1− δ) + (1 + δ)e

−ik

(1− δ) + (1 + δ)eik 0

(16b)

= σx [(1− δ) + (1 + δ) cos k] + σy(1 + δ) sin k, (16c)

where σx and σy are the first two Pauli matrices acting on the sublattice index, t is the

nearest-neighbor hopping integral, and δ is a dimensionless parametrization of the strong-

16

−1.5

−1

−0.5

0

0.5

1

1.5

E

−1.5

−1

−0.5

0

0.5

1

1.5

E

(a) (b)

Figure 3: Energy spectra for the 1D p-wave wire with open boundary con-ditions in the (a) trivial phase (b)non-trivial topological phase with a zeroenergy mode on each boundary point.

18

FIG. 1. Energy spectra for the Su-Schrieffer-Heeger model with open boundary conditions (a)

in the trivial phase and (b) in the nontrivial topological phase with a zero energy mode on each

boundary point.

weak dimerization of bonds.

We observe that Hamiltonian (16) has time-reversal symmetry T = K, chiral symmetryC = σz and thus also particle-hole symmetry P = σzK. This places it in class BDI of Tab. Iwith a Z topological characterization. Observe that breaking the time-reversal symmetry

would not alter the topological properties, as long as the chiral symmetry is intact. The

model would then belong to class AIII, which also features a Z classification. Hence, it is the

chiral symmetry that is crucial to protecting the topological properties of Hamiltonian (16).

Notice that generic longer-range hopping (between sites of the same sublattice) breaks the

chiral symmetry.

What are the different topological sectors that can be accessed by tuning the parameter

δ in the Su-Schrieffer-Heeger model? We observe that the dispersion

ε2k = 2[(1 + δ2) + (1− δ2) cos k

](17)

is gapless for δ = 0, hinting that this is the boundary between two distinct phases δ > 0 and

δ < 0. As we are interested in understanding the topological properties of these phases, we

can analyze them for any convenient value of the parameter δ and then conclude that they

are the same in the entire phase by adiabaticity. We consider the Hamiltonian (15) with

17

open boundary conditions and choose the representative parameters

• δ = +1 : The operators c†1,A and c†N,B do not appear in the Hamiltonian for the openchain. Hence, there exists a state at either end of the open chain that can be occupied

or unoccupied at no cost of energy. Thus, either end of the chain supports a localized

topological end state (see Fig. 1). Away from δ = +1, as long as δ > 0, the end

states start to overlap and split apart in energy by an amount that is exponentially

small in the length N of the chain. We can back up this observation by evaluating the

topological invariant (14) for this phase. The off-diagonal projector is qk = e−ik and

its winding number evaluates to

W(0) =i

2π

∫dk eik(−i)e−ik = 1. (18)

• δ = −1 : In this case strong bonds form between the two sites in every unit cell and notopological end states appear. Correspondingly, as the off-diagonal projector qk = 1

is independent of k, we conclude that the winding number vanishes W(0) = 0.

One can visualize the winding number of a two-band Hamiltonian that has the form

hk = dk · σ in the following way. If the Hamiltonian has chiral symmetry, we can choosethis symmetry to be represented by C = σz without loss of generality. Then dk has to lie in

the x-y-plane for every k and may not be zero if the phase is gapped. The winding number

W(0) measures how often dk winds around the origin in the x-y-plane as k changes from 0

to 2π.

Besides the topological end states, the Su-Schrieffer-Heeger model also features topolog-

ical domain wall states between a region with δ > 0 and δ < 0. Such topological midgap

modes have to appear pairwise in any periodic geometry. As the system is considered at

half filling, each of these modes binds half an electron charge. This is an example of charge

fractionalization at topological defects. It is important to remember that these defects are

not dynamical, but are rigidly fixed external perturbations. Therefore, this form of fraction-

alization is not related to intrinsic topological order.

D. The one-dimensional p-wave superconductor

In the Su-Schrieffer-Heeger model, particle-hole symmetry (and with it the chiral symme-

try) is in some sense fine-tuned, as it is lost if generic longer-range hoppings are considered.

18

In superconductors, particle-hole symmetry arises more naturally as a symmetry that is

inherent in the redundant description of mean-field Bogoliubov-deGennes Hamiltonians.

Here, we want to consider the simplest model for a topological superconductor that has

been studied by Kitaev in Ref. 6. The setup is again a 1D chain with one orbital for spinless

fermion on each site. Superconductivity is encoded in pairing terms c†ic†i+1 that do not

conserve particle number. The Hamiltonian is given by

H =N∑

i=1

[−t(c†ici+1 + c

†i+1ci

)− µc†ici + ∆c†i+1c†i + ∆∗cici+1

]. (19)

Here, µ is the chemical potential and ∆ is the superconducting order parameter, which we

will decompose into its amplitude |∆| and complex phase θ, i.e., ∆ = |∆|eiθ.The fermionic operators c†i obey the algebra

{c†i , cj} = δi,j, (20)

with all other anticommutators vanishing. We can chose to trade the operators c†i and ci on

every site i for two other operators ai and bi that are defined by

ai = e−iθ/2ci + e

iθ/2c†i , bi =1

i

(e−iθ/2ci − eiθ/2c†i

). (21)

These so-called Majorana operators obey the algebra

{ai, aj} = {bi, bj} = 2δij, {ai, bj} = 0 ∀i, j. (22)

In particular, they square to 1

a2i = b2i = 1, (23)

and are self-conjugate

a†i = ai, b†i = bi. (24)

In fact, we can always break up a complex fermion operator on a lattice site into its real

and imaginary Majorana components though it may not always be a useful representation.

As an aside, note that the Majorana anti-commutation relation in Eq. (22) is the same as

that of the generators of a Clifford algebra where the generators all square to +1. Thus,

mathematically one can think of the operators ai (or bi) as matrices forming by themselves

the representation of Clifford algebra generators.

19

1 2

cj

a2j-1 a2j

{

(a)

(b)

Figure 4: Schematic illustration of the lattice p-wave superconductor Hamil-tonian in the (a) trivial limit (b) non-trivial limit. The white (empty) andred(filled) circles represent the Majorana fermions making up each physicalsite (oval). The fermion operator on each physical site (cj) is split up into twoMajorana operators (a2j�1 and a2j). In the non-trivial phase the unpairedMajorana fermion states at the end of the chain are labelled with a 1 and a2. These are the states which are continuously connected to the zero-modesin the non-trivial topological superconductor phase.

though it may not always be a useful representation. As an aside, note thatthe Majorana anti-commutation relation in Eq. 45 is the same as that ofthe generators of a Cli↵ord algebra where the generators all square to +1.Thus, mathematically one can think of the operators ai as matrices formingthe representation of Cli↵ord algebra generators.

Using the Majorana representation the Hamiltonian for the lattice p-wavewire becomes

HBdG =i

2

X

j

(�µa2j�1a2j + (t + |�|)a2ja2j+1 + (�t + |�|)a2j�1a2j+2) . (47)

The factor of i in front of the Hamiltonian may seem out of place, but itis required for Hermiticity when using the Majorana representation. As aquick example, one can see that an operator like (a2ja2j�1)

† = a†2j�1a†2j =

a2j�1a2j = �a2ja2j�1 is anti-Hermitian and becomes Hermitian if a factor ofi is added i.e. ia2ja2j�1 is Hermitian.

In this representation we can illustrate the key di↵erence between thetopological and trivial phases by looking at two special limits

21

|{z}cj

a)

b)

aj bj

1 2

cj

a2j-1 a2j

{

(a)

(b)




HBdG =i

2

X

j



† = a†2j�1a†2j =



21

1 2

cj

a2j-1 a2j

{

(a)

(b)




HBdG =i

2

X

j



† = a†2j�1a†2j =



21

a1 bN

FIG. 2. Schematic illustration of the lattice p-wave superconductor Hamiltonian in the (a) trivial

limit (b) non-trivial limit. The white (empty) and red (filled) circles represent the Majorana

fermions making up each physical site (oval). The fermion operator on each physical site (cj) is

split up into two Majorana operators (aj and bj). In the non-trivial phase the unpaired Majorana

fermion states at the end of the chain are labelled with a1 and bN . These are the states which are

continuously connected to the zero-modes in the non-trivial topological superconductor phase.

When rewritten in the Majorana operators, Hamiltonian (19) takes (up to a constant)

the form

H =i

2

N∑

i=1

[−µai bi + (t+ |∆|)bi ai+1 + (−t+ |∆|)ai bi+1] . (25)

After imposing periodic boundary conditions, it is again convenient to study the system in

momentum space. When defining the Fourier transform of the Majorana operators ai =∑

i eikiak we note that the the self-conjugate property (24) that is local in position space

translates into a†k = a−k in momentum space (and likewise for the bk). The momentum

space representation of the Hamiltonian is

H =∑

k∈BZ

∑

α=A,B

(ak bk)hk

a−kb−k

(26a)

hk =

0 −

iµ2

+ it cos k + |∆| sin kiµ2− it cos k + |∆| sin k 0

(26b)

= σx|∆| sin k + σy(µ

2− t cos k

), (26c)

While this Bloch Hamiltonian is formally very similar to that of the Su-Schrieffer-Heeger

model (16), we have to keep in mind that it acts on entirely different single-particle degrees of

freedom, namely in the space of Majorana operators instead of complex fermionic operators.

20

As with the case of the Su-Schrieffer-Heeger model, the Hamiltonian (26) has a time-reversal

symmetry T = σzK and a particle-hole symmetry P = K which combine to the chiralsymmetry C = σz. Hence, it belongs to symmetry class BDI as well. For the topological

properties that we explore below, only the particle-hole symmetry is crucial. If time-reversal

symmetry is broken, the model changes to symmetry class D, which still supports a Z2topological grading.

To determine its topological phases, we notice that Hamiltonian (26) is gapped except

for |t| = |µ/2|. We specialize again on convenient parameter values on either side of thispotential topological phase transition

• µ = 0, |∆| = t : The Bloch matrix hk takes exactly the same form as that of theSu-Schrieffer-Heeger model (16) for the parameter choice δ = +1. We conclude that

the Hamiltonian (26) is in a topological phase. The Hamiltonian reduces to

H = it∑

j

bjaj+1. (27)

A pictorial representation of this Hamiltonian is shown in Fig. 2 b). With open

boundary conditions it is clear that the Majorana operators a1 and bN are not coupled

to the rest of the chain and are ‘unpaired’. In this limit the existence of two Majorana

zero modes localized on the ends of the chain is manifest.

• ∆ = t = 0, µ < 0 : This is the topologically trivial phase. The Hamiltonian isindependent of k and we conclude that the winding number vanishes W(0) = 0. In this

case the Hamiltonian reduces to

H = −µ i2

∑

j

ajbj. (28)

In its ground state the Majorana operators on each physical site are coupled but the

Majorana operators between each physical site are decoupled. In terms of the physical

complex fermions, it is the ground state with either all sites occupied or all sites empty.

A representation of this Hamiltonian is shown in Fig. 2 a). The Hamiltonian in the

physical-site basis is in the atomic limit, which is another way to see that the ground

state is trivial. If the chain has open boundary conditions there will be no low-energy

states on the end of the chain if the boundaries are cut between physical sites. That

21

is, we are not allowed to pick boundary conditions where a physical complex fermionic

site is cut in half.

These two limits give the simplest representations of the trivial and non-trivial phases.

By tuning away from these limits the Hamiltonian will have some mixture of couplings

between Majorana operators on the same physical site, and operators between physical

sites. However, since the two Majorana modes are localized at different ends of a gapped

chain, the coupling between them will be exponentially small in the length of the wire and

they will remain at zero energy. In fact, in the non-trivial phase the zero modes will not be

destroyed until the bulk gap closes at a critical point.

It is important to note that these zero modes count to a different many-body ground state

degeneracy than the end modes of the Su-Schrieffer-Heeger model. The difference is rooted

in the fact that one cannot build a fermionic Fock space out of an odd number of Majorana

modes, because they are linear combinations of particles and holes. Rather, we can define

a single fermionic operator out of both Majorana end modes a1 and bN as c† := a1 + ibN .

The Hilbert space we can build out of a1 and bN is hence inherently nonlocal. This nonlocal

state can be either occupied or empty giving rise to a two-fold degenerate ground state of

the chain with two open ends. (In contrast, the topological Su-Schrieffer-Heeger chain has

a four-fold degenerate ground state with two open ends, because it has one fermionic mode

on each end.) The Majorana chain thus displays a different form of fractionalization than

the Su-Schrieffer-Heeger chain. For the latter, we observed that the topological end modes

carry fractional charge. In the Majorana chain, the end modes are a fractionalization of a

fermionic mode into a superposition of particle and hole (and have no well defined charge

anymore), but the states |0〉 (with c|0〉 = 0) and c†|0〉 do have distinct fermion parity. Thenonlocal fermionic mode formed by two Majorana end modes is envisioned to work as a

qubit (a quantum-mechanical two-level system) that stores quantum information (its state)

in a way that is protected against local noise and decoherence.

E. Reduction of the 10-fold way classification by interactions: Z→ Z8 in class BDI

When time-reversal symmetry T = K is present, the model considered in Sec. I D belongsto class BDI of the classification of noninteracting fermionic Hamiltonians in Tab. I with a

Z topological characterization. We want to explore how interactions alter this classification,

22

following a calculation by Fidkowski and Kitaev from Ref. 8. To this end, we consider a

collection of n identical 1D topological Majorana chains in class BDI and only consider their

Majorana end modes on one end, which we denote by γ1, · · · , γn. We will take the point ofview that if we can gap the edge, we can continue the bulk to a trivial state (insulator). This

is not entirely a correct point of view in general (see 2D topologically ordered states such

as the toric code discussed in the next Section), but works for our purposes. Given some

integer n, we ask whether we can couple the Majorana modes locally on one end such that

no gapless degrees of freedom are left on that end and the ground state with open boundary

conditions becomes singly degenerate. To remain in class BDI, we only allow couplings that

respect time-reversal symmetry. Let us first derive the action of T on the Majorana modes.The complex fermion operators are left invariant under time-reversal T cT −1 = c. Hence,

T (a+ ib)T −1 = T aT −1 − iT bT −1 != a+ ib ⇒ T aT −1 = a, T bT −1 = −b. (29)

Thus, when acting on the modes localized on the left end of the wire (which transform like

the a’s), time-reversal symmetry leaves the Majorana operators invariant.

The most naive coupling term that would gap out two Majoranas is iγ1γ2. This is because

two Majoranas can form a local Hilbert space (unlike just one Majorana), and this local

Hilbert space can be split unless some other symmetry prevents it from being split. However,

time-reversal symmetry forbids these hybridization terms, for it sends iγ1γ2 → −iγ1γ2. Inspinful systems, another symmetry which can do this is MT , where M is a mirror operator(which in spinful systems squares to −1 M2 = −1) and T is the usual time-reversal operatorT 2 = −1, such that (MT )2 = M2T 2 = 1 and hence MT acts like spinless time reversal.7

Realizing that such a term is not allowed is the end of the story for noninteracting systems,

yielding the classification Z. Lets find out what interactions do to this system. The steps

that we will now outline are summarized in Fig. 3.

We saw that two Majorana end states cannot be gapped: the only possible interacting

or noninteracting Hamiltonian is ia1a2. Three Majoranas clearly cannot be gapped either,

as it is an odd number. Let us thus add two more Majorana end states into the mix. Any

one-body term still is disallowed but the term

Hint = a1a2a3a4 (30)

can be present. We can now form two complex fermions, c1 = (a1 + ia2)/√

2, c2 = (a3 +

23

Hint H(2)int

2 wires 4 wires 8 wires

FIG. 3. Schematic illustration of the many body energy levels for 2, 4, and 8 wires with Majorana

end states as well as the (partial) lifting of their degeneracy by the Hamiltonians in Eqs. (31)

and (32).

ia4)/√

2. In terms of these two fermions, the Hamiltonian reads

Hint = −(n1 −

1

2

)(n2 −

1

2

), (31)

where n1 = c†1c1 and n2 = c

†2c2 are the occupation numbers. The Hamiltonian is diagonal

in the eigenbasis |n1n2〉 of the occupation number operators, and the states |11〉 , |00〉 aredegenerate at energy −1/4, while the states |01〉 , |10〉 are degenerate at energy +1/4. Theoriginal noninteracting system of four Majorana fermions had a degeneracy of 22 = 4. The

interaction, however, has lifted this degeneracy, but not all the way to a single nondegen-

erate ground state. Irrespective of the sign of the interaction, it leaves the states doubly

degenerate on one edge, and hence cannot be adiabatically continued to the trivial state of

single degeneracy. However, if we add four more Majoranas wires so that we have n = 8

Majoranas, we can build an interaction which creates a singly degenerate ground state. We

can understand this as follows: Add two interactions

H(1)int = −α(a1a2a3a4 + a5a6a7a8) (32a)

These create two doublets, one in c1, c2 defined above, and one in c3 = (a5 + ia6)/√

2, c4 =

(a7 + ia8)/√

2. We couple these doublets via the interaction

H(2)int =

∑

i=x,y,z

β(c†1 c†2)σi

c1c2

(c†3 c†4)σi

c3c4

. (32b)

Representing each of the doublets as a spin-1/2 S, this interaction is nothing but an S · Sterm. If we take 0 < β � α, then we can approximate the interaction β by its action within

24

the two ground state doublets. As such, this interaction creates a singlet and a triplet (in

that doublet) and for the right sign of β, we can put the singlet below the triplet, thereby

creating a unique ground state

1√2

(|0110〉 − |1001〉) , (33)

in terms of the occupation number states |n1n2n3n4〉. This unique ground state can beadiabatically continued to the atomic limit. In this way the noninteracting Z classification

of class BDI breaks down to Z8 if interactions are allowed.

25

II. EXAMPLES OF TOPOLOGICAL ORDER

So far, we have been concerned with symmetry protected topological states and consid-

ered examples that were motivated by the topological classification of free fermion Hamilto-

nians. The topological properties of these systems are manifest by the presence of protected

boundary modes.

In this Section, we want to familiarize ourselves with the concept of intrinsic topological

order by ways of two examples. We will study the connections between different characteri-

zations of topological order, such as fractionalized excitations in the bulk and the topological

ground state degeneracy. Our examples will be in 2D space, as topologically ordered states

do not exist in 1D and are best understood in 2D. Our first example, the toric code, has

Abelian anyon excitations, while the second example, the chiral p-wave superconductor,

features non-Abelian anyons.

A. The toric code

The first example of a topologically ordered state is an exactly soluble model with van-

ishing correlation length. The significance of having zero correlation length is the following.

The correlation functions of local operators decay exponentially in gapped quantum ground

states in 1D and 2D with a characteristic length scale given by the correlation length ξ.9

In contrast, topological properties are encoded in quantized expectation values of nonlocal

operators (for example the Hall conductivity) or the degeneracy of energy levels (such as

the end states of the Su-Schrieffer-Heeger model). In finite systems, such quantizations and

degeneracies are generically only exact up to corrections that are of order e−L/ξ, where L

is the linear system size. Models with zero correlation length are free from such exponen-

tial finite-size corrections and thus expose the topological features already for the smallest

possible system sizes. The down-side is that their Hamiltonians are rather contrived.

We define the toric code model10 on a square lattice with a spin-1/2 degree of freedom on

every bond j (see Fig. 4). The four spins that sit on the bonds emanating from a given site

of the lattice are referred to as a star s. The four spins that sit on the bonds surrounding a

26

As

Bp

FIG. 4. The toric code model is defined on a square lattice with spin-1/2 degrees of freedom on

every bond (black squares). The operator As acts with σx on all four spins one the bonds that

are connected to a lattice site (a star s). The operator Bp acts with σz on all four spins around a

plaquette p.

square of the lattice are called a plaquette p. We define two sets of operators

As :=∏

j∈sσxj , Bp :=

∏

j∈pσzj , (34)

that act on the spins of a given star s and plaquette p, respectively. Here, σx,zj are the

respective Pauli matrices acting on the spin on bond j.

These operators have two crucial properties which are often used to construct exactly

soluble models for topological states of matter

1. All of the As and Bp commute with each other. This is trivial for all cases except

for the commutator of As with Bp if s and p have spins in common. However, any

star shares with any plaquette an even number of spins (edges), so that commuting

As with Bp involves commuting an even number of σz with σx, each of which comes

with a minus sign.

2. The operators

1−Bp2

,1− As

2(35)

are projectors. The former projects out plaquette states with an even number of spins

polarized in the positive z-direction. The latter projects out stars with an even number

of spins in the positive x-direction.

27

1. Ground states

The Hamiltonian is defined as a sum over these commuting projectors

H = −Je∑

s

As − Jm∑

p

Bp, (36)

where the sums run over all stars s and plaquettes p of the lattice. Let us assume that both

Je and Jm are positive constants. Then, the ground state is given by a state in which all

stars s and plaquettes p are in an eigenstate with eigenvalue +1 of As and Bp, respectively.

(The fact that all As and Bp commute allows for such a state to exist, as we can diagonalize

each of them separately.) Let us think about the ground state in the eigenbasis of the σx

operators and represent by bold lines those bonds with spin up and and draw no lines along

bonds with spin down. Then, As imposes on all spin configurations with nonzero amplitude

in the ground state the constraint that an even number of bold lines meets at the star s.

In other words, we can think of the bold lines as connected across the lattice and they may

only form closed loops. Bold lines that end at some star (“open strings”) are not allowed

in the ground state configurations; they are excited states. Having found out which spin

configurations are allowed in the ground state, we need to determine their amplitudes. This

can be inferred from the action of the Bp operators on these closed loop configurations. The

Bp flips all bonds around the plaquette p. Since B2p = 1, given a spin configuration |c〉 in

the σx-basis, we can write an eigenstate of Bp with eigenvalue 1 as

1√2

(|c〉+Bp|c〉) , (37)

for some fixed p. This reasoning can be extended to all plaquettes so that we can write for

the ground state

|GS〉 =(∏

p

1 +Bp√2

)|c〉, (38)

where |c〉 is a closed loop configuration [see Fig. 5 a)]. Is |GS〉 independent of the choice of|c〉? In other words, in the ground state unique? We will see that the answer depends onthe topological properties of the manifold on which the lattice is defined and thus reveals

the topological order imprinted in |GS〉.To answer these questions, let us consider the system on two topologically distinct mani-

folds, the torus and the sphere. To obtain a torus, we consider a square lattice with Lx×Ly

28

|GSi = + + + · · ·

a)

b) c) d) e)

FIG. 5. Visualization of the toric code ground states on the torus. a) The toric code ground state is

the equal amplitude superposition of all closed loop configurations. b)-e) Four base configurations

|c〉 entering Eq. (38) that yield topologically distinct ground states on the torus.

sites and impose periodic boundary conditions. This lattice hosts 2LxLy spins (2 per unit

cell for they are centered along the bonds). Thus, the Hilbert space of the model has di-

mension 22LxLy . There are LxLy operators As and just as many Bp. Hence, together they

impose 2LxLy constraints on the ground state in this Hilbert space. However, not all of

these constraints are independent. The relations

1 =∏

s

As, 1 =∏

p

Bp (39)

make two of the constraints redundant, yielding (2LxLy − 2) independent constraints. Theground state degeneracy (GSD) is obtained as the quotient of the Hilbert space dimension

and the subspace modded out by the constraints

GSD =22LxLy

22LxLy−2= 4. (40)

The four ground states on the torus are distinguished by having an even or an odd number

of loops wrapping the torus in the x and y direction, respectively. Four configurations |c〉that can be used to build the four degenerate ground states are shown in Fig. 5 b)-e).

This constitutes a set of “topologically degenerate” ground states and is a hallmark of the

topological order in the model.

Let us contrast this with the ground state degeneracy on the sphere. Since we use a

zero correlation length model, we might as well use the smallest convenient lattice with the

29

topology of a sphere. We consider the model (36) defined on the edges of a cube. The

same counting as above yields that there are 12 degrees of freedom (the spins on the 12

edges), 8 constraints from the As operators defined on the corners and 6 constraints from

the Bp operators defined on the faces. Subtracting the 2 redundant constraints (39) yields

12− (8 + 6− 2) = 0 remaining degrees of freedom. Hence, the model has a unique groundstate on the sphere.

On a general manifold, we have

GSD = 2number of noncontractible loops. (41)

An important property of the topologically degenerate ground states is that any local oper-

ator has vanishing off-diagonal matrix elements between them in the thermodynamic limit.

Similarly, no local operator can be used to distinguish between the ground states. We can,

however, define nonlocal operators that transform one topologically degenerate ground state

into another and that distinguish the ground states by topological quantum numbers. (No-

tice that such operators may not appear in any physical Hamiltonian due to their nonlocality

and hence the degeneracy of the ground states is protected.) On the torus, we define two

pairs of so-called Wilson loop operators as

W ex/y :=∏

j∈lex/y

σzj , Wmx/y :=

∏

j∈lmx/y

σxj . (42)

Here, lex/y are the sets of spins on bonds parallel to a straight line wrapping the torus

once along the x- and y-direction, respectively. The lmx/y are the sets of spins on bonds

perpendicular to a straight line that connects the centers of plaquettes and wraps the torus

once along the x and y-direction, respectively. We note that the W ex/y and Wmx/y commute

with all As and Bp [W

e/mx/y , As

]=[W

e/mx/y , Bp

]= 0, (43)

and thus also with the Hamiltonian. Furthermore, they obey

W exWmy = −Wmy W ex . (44)

This algebra must be realized in any eigenspace of the Hamiltonian. However, due to

Eq. (44), it cannot be realized in a one-dimensional subspace. We conclude that all

eigenspaces of the Hamiltonian, including the ground state, must be degenerate. In the

30

a) b)

c) d)

e1 e2m2m1

e1e2 m

em

e

FIG. 6. Visualization of operations to compute the braiding statistics of toric code anyons. a)

Two e excitations above the ground state. b) Two m excitations above the ground state. c) Loop

created by braiding e1 around e2. c) Loop created by braiding e around m. A phase of −1 results

for this process because there is a single bond on which both a σx operator (dotted line) and a σz

operator (bold line) act.

σx basis that we used above, Wmx/y measures whether the number of loops wrapping the

torus is even or odd in the x and y direction, respectively, giving 4 degenerate ground states.

In contrast, W ex/y changes the number of loops wrapping the torus in the x and y direction

between even and odd.

2. Topological excitations

To find the topological excitations of the system above the ground state, we ask which

are the lowest energy excitations that we can build. Excitations are a violation of the rule

that all stars s are eigenstates of As and all plaquettes p are eigenstates of Bp. Let us first

focus on star excitations which we will call e. They appear as the end point of open strings,

i.e., if the closed loop condition is violated. Since any string has two end points, the lowest

excitation of this type is a pair of e. They can be created by acting on the ground state

with the operator

W ele :=∏

j∈leσzj , (45)

where le is a string of bonds connecting the two excitations e1 and e2 [see Fig. 6 a)]. The

state

|e1, e2〉 := W ele|GS〉 (46)

31

has energy 4Je above the ground state energy. Similarly, we can define an operator

Wmlm :=∏

j∈lmσxj , (47)

that creates a pair of plaquette defectsm1 andm2 connected by the string lm of perpendicular

bonds [see Fig. 6 b)]. (Notice that the operator Wmlm does not flip spins when the ground state

is written in the σx basis. Rather, it gives weight +1/−1 to the different loop configurationsin the ground state, depending on whether an even or an odd number of loops crosses lm.)

The state

|m1,m2〉 := Wmlm|GS〉 (48)

has energy 4Jm above the ground state energy. Notice that the excited states |e1, e2〉 and|m1,m2〉 only depend on the positions of the excitations and not on the particular choice ofstring that connects them. Furthermore, the energy of the excited state is independent of

the separation between the excitations. The excitations are thus “deconfined”, i.e., free to

move independent of each other.

It is also possible to create a combined defect when a plaquette hosts a m excitation and

one of its corners hosts a e excitation. We call this combined defect f and formalize the

relation between these defects in a so-called fusion rule

e×m = f. (49a)

When two e-type excitations are moved to the same star, the loop le that connects them

becomes a closed loop and the state returns to the ground state. For this, we write the

fusion rule

e× e = 1, (49b)

where 1 stands for the ground state or vacuum. Similarly, moving two m-type excitations

to the same plaquette creates a closed loop lm, which can be absorbed in the ground state,

i.e.,

m×m = 1. (49c)

Superimposing the above processes yields the remaining fusion rules

m× f = e, e× f = m, f × f = 1. (49d)

It is now imperative to ask what type of quantum statistics these emergent excitations

obey. We recall that quantum statistics are defined as the phase by which a state changes if

32

two identical particles are exchanged. Rendering the exchange operation as an adiabatically

slow evolution of the state, in three and higher dimensions only two types of statistics are

allowed between point particles: that of bosons with phase +1 and that of fermions with

phase −1. In 2D, richer possibilities exist and the exchange phase θ can be any complexnumber on the unit circle, opening the way for anyons. While the exchange is only defined for

quantum particles of the same type, the double exchange (braiding) is well defined between

any two deconfined anyons. We can compute the braiding phases of the anyons e, m, and f

that appear in the toric code one by one. Let us start with the phase resulting from braiding

e1 with e2. The initial state is Wele|GS〉 depicted in Fig. 6 a). Moving e1 around e2 leaves a

loop of flipped σx bonds around e2 [see Fig. 6 c)]. This loop is created by applying Bp to all

plaquettes enclosed by the loop lee1 along which e1 moves. We can thus write the final state

as∏

p∈lee1

Bp

W ele|GS〉 =W ele

∏

p∈lee1

Bp

|GS〉

=W ele|GS〉.

(50)

Flipping the spins in a closed loop does not alter the ground state as it is the equal amplitude

of all loop configurations. We conclude that the braiding of two e particles gives no phase.

Similar considerations can be used to conclude that the braiding of two m particles is trivial

as well. In fact, not only the braiding, but also the exchange of two e particles and two m

particles is trivial. (We have not shown that here.)

More interesting is the braiding of m with e. Let the initial state be WmlmWele|GS〉 and

move the e particle located on one end of the string lein around the magnetic particle m on

one end of the string lm. Again this is equivalent to applying Bp to all plaquettes enclosed

by the path lee of the e particle, so that the final state is given by∏

p∈lee

Bp

WmlmW ele|GS〉 = −Wmlm

∏

p∈lee

Bp

W ele|GS〉

= −WmlmW ele|GS〉.

(51)

The product over Bp operators anticommutes with the path operator Wmlm , because there

is a single bond on which a single σx and a single σz act at the crossing of lm and lee [see

Fig. 6 d)]. As a result, the initial and final state differ by a −1, which is the braiding phaseof e with m. Particles with this braiding phase are called (mutual) semions.

33

Notice that we have moved the particles on contractible loops only. If we create a pair of e

orm particles, move one of them along a noncontractible loop on the torus, and annihilate the

pair, we have effectively applied the operators W ex/y and Wmx/y to the ground state (although

in the process we have created finite energy states). The operation of moving anyons on

noncontractible loops thus allows to operate on the manifold of topologically degenerate

groundstates. This exposes the intimate connection between the presence of fractionalized

excitations and topological groundstate degeneracy in topologically ordered systems.

From the braiding relations of e and m we can also conclude the braiding and exchange

relations of the composite particle f . This is most easily done in a pictorial way by repre-

senting the particle worldlines as moving upwards. For example, we represent the braiding

relations of e and m as

e e e

=

e

=

m m m m

tim

e

e m e m

= � . (52)

The exchange of two f , each of which is composed of one e and one m is then

e m eme m em

=

e m

= �

m e|{z}f

|{z}f

(53)

Notice that we have used Eq. (52) to manipulate the crossing in the dotted rectangles.

Exchange of two f thus gives a phase −1 and we conclude that f is a fermion.In summary, we have used the toric code model to illustrate topological ground state

degeneracy and emergent anyonic quasiparticles as hallmarks of topological order. We note

that the toric code model does not support topologically protected edge states.

B. The two-dimensional p-wave superconductor

The second example of a 2D system with anyonic excitations that we want to discuss

here is the chiral p-wave superconductor. Unlike the toric code, due to its chiral nature, it is

34

a model with nonzero correlation length. The vortices of the chiral p-wave superconductor

exhibit anyon excitations which have exotic non-Abelian statistics.12–14 (The anyons in the

toric code are Abelian, we will see below what that distinction refers to.) For the system to

be topologically ordered, these vortices should appear as emergent, dynamical excitations.

This requires to treat the electromagnetic gauge field quantum-mechanically. (In fact, since

the fermion number conservation is spontaneously broken down to the conservation of the

fermion parity in the superconductor, the relevant gauge theory involves only a Z2 instead

of a U(1) gauge field.) However, the topological properties that we want to discuss here

can also be seen if we model the gauge field and vortices as static defects, rather than

within a fluctuating Z2 gauge theory. This allows us to study a models very similar to

the “noninteracting” topological superconductor in 1D and still expose the non-Abelian

statistics.

For pedagogy we will use both lattice and continuum models of the chiral superconductor.

We begin with the lattice Hamiltonian defined on a square lattice

H =∑

m,n

{−t(c†m+1,ncm,n + c

†m,n+1cm,n + h.c.

)− (µ− 4t)c†m,ncm,n

+(

∆c†m+1,nc†m,n + i∆c

†m,n+1c

†m,n + h.c.

)}.

(54)

The fermion operators cm,n annihilate fermions on the lattice site (m,n) and we are consid-

ering spinless (or equivalently spin-polarized) fermions. We set the lattice constant a = 1

for simplicity. The pairing amplitude is anisotropic and has an additional phase of i in the

y-direction compared to the pairing in the x-direction. Because the pairing is not on-site,

just as in the lattice version of the p-wave wire, the pairing terms will have momentum de-

pendence. We can write this Hamiltonian in the Bogoliubov-deGennes form and, assuming

that ∆ is translationally invariant, can Fourier transform the lattice model to get

HBdG =1

2

∑

p

Ψ†p

�(p) 2i∆(sin px + i sin py)−2i∆∗(sin px − i sin py) −�(p)

Ψp, (55)

where �(p) = −2t(cos px + cos py)− (µ− 4t) and Ψp =(cp c

†−p

)T. For convenience we have

shifted the chemical potential by the constant 4t. As a quick aside we note that the model

takes a simple familiar form in the continuum limit (p→ 0):

H(cont)BdG =

1

2

∑

p

Ψ†p

p2

2m− µ 2i∆(px + ipy)

−2i∆∗(px − ipy) − p2

2m+ µ

Ψp (56)

35

where m ≡ 1/2t and p2 = p2x + p2y. We see that the continuum limit has the characteristicpx + ipy chiral form for the pairing potential. The quasiparticle spectrum of H

(cont)BdG is

E± = ±√

(p2/2m− µ)2 + 4|∆|2p2, which, with a nonvanishing pairing amplitude, is gappedacross the entire BZ as long as µ 6= 0. This is unlike some other types of p-wave pairingterms [e.g., ∆(p) = ∆px] which can have gapless nodal points or lines in the BZ for µ > 0.

In fact, nodal superconductors, having gapless quasiparticle spectra, are not topological

superconductors by definition (i.e., a bulk excitation gap does not exist).

We recognize the form of H(cont)BdG as a massive 2D Dirac Hamiltonian, and indeed Eq. (54)

is just a lattice Dirac Hamiltonian which is what we will consider first. In the first quantized

notation, the single particle Hamiltonian for a superconductor is equivalent to that of an

insulator with an additional particle-hole symmetry. It is thus placed in class D of Tab. I

and admits a Z topological classification in 2D. Thus, we can classify the eigenstates of

Hamiltonian (54) by a Chern number – but due to the breaking of U(1) symmetry, the

Chern number does not have the interpretation of Hall conductance. However, it is still a

topological invariant.

We expect that HBdG will exhibit several phases as a function of ∆ and µ for a fixed t > 0.

For simplicity let us set t = 1/2 and make a gauge transformation cp → eiθ/2cp, c†p → e−iθ/2c†pwhere ∆ = |∆|eiθ. The Bloch Hamiltonian for the lattice superconductor is then

HBdG(p) = (2− µ− cos px − cos py)σz − 2|∆| sin pxσy − 2|∆| sin pyσx, (57)

where the σi, i = x, y, z, are the Pauli matrices in the particle/hole basis. Assuming

|∆| 6= 0, this Hamiltonian has several fully-gapped superconducting phases separated bygapless critical points. The quasi-particle spectrum for the lattice model is

E± = ±√

(2− µ− cos px − cos py)2 + 4|∆|2 sin2 px + 4|∆|2 sin2 py (58)

and is gapped (under the assumption that |∆| 6= 0) unless the prefactors of all three Paulimatrices vanish simultaneously. As a function of (px, py, µ) we find three critical points.

The first critical point occurs at (px, py, µ) = (0, 0, 0). The second critical point has two

gap-closings in the BZ for the same value of µ : (π, 0, 2) and (0, π, 2). The third critical

point is again a singly degenerate point at (π, π, 4). We will show that the phases for µ < 0

and µ > 4 are trivial superconductors while the phases 0 < µ < 2 and 2 < µ < 4 are

topological superconductors with opposite chirality. In principle one can define a Chern

36

number topological invariant constructed from the eigenstates of the lower quasi-particle

band to characterize the phases. We will show this calculation below, but first we make

some physical arguments as to the nature of the phases, following the discussion in Ref. 11.

We will first consider the phase transition at µ = 0. The low-energy physics for this

transition occurs around (px, py) = (0, 0) and so we can expand the lattice Hamiltonian

around this point; this is nothing but Eq. (56). One way to test the character of the µ < 0

and µ > 0 phases is to make an interface between them. If we can find a continuous

interpolation between these two regimes which is always gapped then they are topologically

equivalent phases of matter. If we cannot find such a continuously gapped interpolation

then they are topologically distinct. A simple geometry to study is a domain wall where

µ = µ(x) such that µ(x) = −µ0 for x < 0 and µ(x) = +µ0 for x > 0 for a positive constantµ0. This is an interface which is translationally invariant along the y-direction, and thus we

can consider the momentum py as a good quantum number to simplify the calculation. What

we will now show is that there exist gapless, propagating fermions bound to the interface

which prevent us from continuously connecting the µ < 0 phase to the µ > 0 phase. This is

one indication that the two phases represent topologically distinct classes.

The single-particle Hamiltonian in this geometry is

HBdG(py) =1

2

−µ(x) 2i|∆|

(−i d

dx+ ipy

)

−2i|∆|(−i d

dx− ipy

)µ(x)

, (59)

where we have ignored the quadratic terms in p, and py is a constant parameter, not an op-

erator. This is a quasi-1D Hamiltonian that can be solved for each value of py independently.

We propose an ansatz for the gapless interface states:

|ψpy(x, y)〉 = eipyy exp(− 1

2|∆|

∫ x

0

µ(x′)dx′)|φ0〉 (60)

for a constant, normalized spinor |φ0〉. The secular equation for a zero-energy mode at py = 0is

HBdG(py)|ψ0(x, y)〉 = 0 =⇒

−µ(x) −µ(x)

µ(x) µ(x)

|φ0〉 = 0. (61)

The constant spinor which is a solution of this equation is |φ0〉 = 1/√

2 (1,−1)T . This formof the constant spinor immediately simplifies the solution of the problem at finite py. We see

that the term proportional to py in Eq. (59) is −2|∆|pyσx. Since σx|φ0〉 = −|φ0〉, i.e., thesolution |φ0〉 is an eigenstate of σx, we conclude that |ψpy(x, y)〉 is an eigenstate of HBdG(py)

37

with energy E(py) = −2|∆|py. Thus, we have found a normalizable bound state solution atthe interface of two regions with µ < 0 and µ > 0 respectively. This set of bound states,

parameterized by the conserved quantum number py is gapless and chiral, i.e., the group

velocity of the quasiparticle dispersion is always negative and never changes sign (in this

simplified model). The chirality is determined by the sign of the “spectral” Chern number

mentioned above which we will calculate below.

These gapless edge states have quite remarkable properties and are not the same chiral

complex fermions that propagate on the edge of integer quantum Hall states, but chiral real

(Majorana) fermions. Using Clifford algebra representation theory it can be shown that

the so-called chiral Majorana (or Majorana-Weyl) fermions can only be found in spacetime

dimensions (8k+ 2), where k = 0, 1, 2, · · · . Thus, we can only find chiral-Majorana states in(1 + 1) dimensions or in (9 + 1) dimensions (or higher!). In condensed matter, we are stuck

with (1 + 1) dimensions where we have now seen that they appear as the boundary states

of chiral topological superconductors. The simplest interpretation of such chiral Majorana

fermions is as half of a conventional chiral fermion, i.e., its real or imaginary part. To show

this, we will consider the edge state of a Chern number 1 quantum Hall system for a single

edge

H(QH)edge = ~v∑

p

p η†pηp, (62)

where p is the momentum along the edge. The fermion operators satisfy{η†p, ηp′

}= δpp′ .

Similar to the discussion on the 1D superconducting wire we can decompose these operators

into their real and imaginary Majorana parts

ηp =1

2(γ1,p + iγ2,p), η

†p =

1

2(γ1,−p − iγ2,−p), (63)

where γa,p (a = 1, 2) are Majorana fermion operators satisfying γ†a,p = γa,−p and

{γa,−p, γb,p′

}=

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2δabδpp′ . The quantum Hall edge Hamiltonian now becomes

H(QH)edge =~v∑

p≥0p(η†pηp − η†−pη−p)

=~v4

∑

p≥0p {(γ1,−p − iγ2,−p)(γ1,p + iγ2,p)− (γ1,p − iγ2,p)(γ1,−p + iγ2,−p)}

=~v4

∑

p≥0p (γ1,−pγ1,p + γ2,−pγ2,p − γ1,pγ1,−p − γ2,pγ2,−p)

=~v2

∑

p≥0p (γ1,−pγ1,p + γ2,−pγ2,p − 2) .

(64)

Thus

H(QH)edge =~v2

∑

p≥0p (γ1,−pγ1,p + γ2,−pγ2,p) (65)

up to a constant shift of the energy. This Hamiltonian is exactly two copies of a chiral

Majorana Hamiltonian. The edge/domain-wall fermion Hamiltonian of the chiral p-wave

superconductor will be

H(p−wave)edge =~v2

∑

p≥0pγ−pγp. (66)

Finding gapless states on a domain wall of µ is an indicator that the phases with µ > 0

and µ < 0 are distinct. If they were the same phase of matter we should be able to

adiabatically connect these states continuously. However, we have shown a specific case of

the more general result that any interface between a region with µ > 0 and a region with

µ < 0 will have gapless states that generate a discontinuity in the interpolation between

the two regions. The question remaining is: Is µ > 0 or µ < 0 non-trivial? The answer is

that we have a trivial superconductor for µ < 0 (adiabatically continued to µ → −∞) anda topological superconductor for µ > 0. Remember that for now we are only considering µ

in the neighborhood of 0 and using the continuum model expanded around (px, py) = (0, 0).

We will now define a bulk topological invariant for 2D superconductors that can distinguish

the trivial superconductor state from the chiral topological superconductor state. For the

spinless Bogoliubov-deGennes Hamiltonian, which is of the form

HBdG =1

2

∑

p

Ψ†p [d(p, µ) · σ] Ψp, (67a)

d(p, µ) =(−2|∆|py,−2|∆|px, p2/2m− µ

), (67b)

the topological invariant is the spectral Chern number defined in Eq. (11), which simplifies,

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for this Hamiltonian, to the winding number

C(1) =1

8π

∫d2p �ij d̂ ·

(∂pid̂× ∂pj d̂

)=

1

8π

∫d2p

�ij

|d|3 d ·(∂pid× ∂pjd

). (68)

We defined the unit vector d̂ = d/

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